Quantum Physics

A photon is a discrete packet {or quantum} of energy of an electromagnetic radiation/wave.

Energy of a photon, E = h f = hc / λ where h: Planck's constant

λviolet ≈ 4 x 10-7 m, λred ≈ 7 x 10-7 m

Power of electromagnetic radiation, P = Rate of incidence of photon x Energy of a photon = (N/t)(hc/λ)

Photoelectric effect refers to the emission of electrons from a cold metal surface when electromagnetic radiation of sufficiently high frequency falls on it.

4 Major Observations:

  1. No electrons are emitted if the frequency of the light is below a minimum frequency {called the threshold frequency}, regardless of the intensity of light
  2. Rate of electron emission {ie photoelectric current} is proportional to the light intensity.
  3. {Emitted electrons have a range of kinetic energy, ranging from zero to a certain maximum value. Increasing the freq increases the kinetic energies of the emitted electrons and in particular, increases the maximum kinetic energy.} This maximum kinetic energy depends only on the frequency and the metal used {ϕ}; the intensity has no effect on the kinetic energy of the electrons.
  4. Emission of electrons begins instantaneously {i.e. no time lag between emission & illumination} even if the intensity is very low.
    NB: (1), (3) & (4) cannot be explained by Wave Theory of Light; instead they provide evidence for the particulate/particle nature of electromagnetic radiation.

Explanation for how photoelectric effect provides evidence for the particulate nature of em radiation:
{Consider the observations (1), (3) & (4). Use any 2 observations above to describe how they provide evidence that em radiation has a particle nature.}

Threshold frequency is the minimum frequency of the em radiation required to eject an electron from a metal surface. {This is because the electrons are held back by the attractive forces of the positive nuclei in the metal.}

Work function of a metal is the minimum energy required to eject an electron from a metal surface

ϕ = h f0 = hcλ0 f0 = threshold frequency
λ0 = threshold wavelength

Maximum KE of electrons, ½ mev2max = eVs {in magnitude} , Vs: stopping potential
hf = ϕ + eVs

From eVs = hf - ϕ

Intensity = Incident Power / Illuminated Area = (N/t)(hc/λ)(1/Area)

Thereby Intensity ∝ Rate of incidence of photons, N/t {for a given λ}

Photocurrent I = (n/t)e, where (n/t) = rate of emission of electrons

Why rate of emission of electrons << rate of incidence of photons {for f>f0}:

1 eV = (1.6 x 10-19C)x (1V) = 1.6 x 10-19J {Using W = QV}
1 nanometre (nm) = 1 x 10-9m

Photoelectric equation: Energy of photon = Work function (energy) + Max. KE of electrons

hf = ϕ + ½ mev2max

Wave-Particle Duality Concept

Electron diffraction provides evidence that matter / particles have also a wave nature & thus, have a dual nature.

de Broglie wavelength of a particle {“matter waves”}, λ = hp

Energy Levels of Isolated Atom:

Explain how existence of electron energy levels in atoms gives rise to line spectra:

2 common ways to cause Excitation of an atom:

The energy level of the ground state gives the ionization energy, i.e. the energy needed to completely remove an electron initially in the ground state from the atom {i.e. to the energy level n = ∞, where E =0}.

Emission line spectrum: A series of discrete/separate bright lines on a dark background, produced by electron transitions within an atom from higher to lower energy levels and emitting photons.

An excited atom during a downward transition emits a photon of frequency f, such that Ei - Ef = hf

Absorption line spectrum: A continuous bright spectrum crossed by “dark” lines. It is produced when “white light” passes through a cool gas. Atoms/electrons of the cool gas absorb photons of certain frequencies and get excited to higher energy levels which are then quickly re-emitted in all directions.

Characteristic X-rays: produced when an electron is knocked out of an inner shell of a target metal atom, allowing another electron from a higher energy level to drop down to fill the vacancy. The x-rays emitted have specific wavelengths, determined by the discrete energy levels which are characteristic of the target atom.

Continuous X-ray Spectrum {Braking Radiation (Bremsstrahlung)}: produced when electrons are suddenly decelerated upon collision with atoms of the metal target.

Minimum λ of cont. spectrum λmin: given by hc / λmin = eVa , Va: accelerating pd of x-ray tube

Heisenberg Uncertainty Principles: If a measurement of the position of a particle is made with uncertainty Δx and a simultaneous measurement of its momentum is made with uncertainty Δp, the product of these 2 uncertainties can never be smaller than h/4π

Δx Δp ≥ h / 4π

Similarly ΔE Δt ≥ h/4π where E is the energy of a particle at time t

A particle can be described by a wave function Ψ where the square of the amplitude of wave function, IΨ I2, is proportional to the probability of finding the particle at a point.

Potential barrier: A region of electric field that prevents an atomic particle like an electron on one side of the barrier from passing through to the other side.


Quantum tunnelling: A quantum-mechanical process whereby a particle penetrates a classically forbidden region of space, i.e. the particle goes through a potential barrier even though it does not have enough energy to overcome it. Due to the wave nature of a particle, there is a non-zero probability that the particle is able to penetrate the potential barrier.

Scanning tunnelling microscope: Involves passing electrons from the tip of a probe through a potential barrier to a material that is to be scanned.

A feedback device adjusts the vertical height of the tip to keep the tunnelling current const as the tip is scanned over the surface {Method 2}). The output of the device provides an image of the surface contour of the material. )

Transmission coefficient (T): measures the probability of a particle tunnelling through a barrier.

T = e -2 k d k = √[(8π2m(U - E))h2]
d: the thickness of the barrier in metres
m: mass of the tunnelling particle in kg
U: the “height” of the potential barrier in J {NOT: eV}
E: the energy of the electron in J
h: The Planck's constant

Reflection coefficient (R): measures the probability that a particle gets reflected by a barrier.

T + R = 1